In Chapter 7, we reviewed the theory of electrodiffusion as applied to a membrane traversed by a steady current of a single species of ions. Let us now consider two complications: multi-ion electrodiffusion and time-dependent electrodiffusion. We will also note that the system of electrodiffusion equations obeys a set of scaling rules. We close this chapter with a critique of the classical electrodiffusion model as applied to membranes containing voltage-sensitive ion channels.
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Notes And References
The two-charge case was analyzed by L. Bass, Trans. Farad. Soc. 60:1656–1663, 1964 and L. J. Bruner, Biophys. J. 5:867–886, 1965. H. R. Leuchtag, J. Math. Physics 22:1317–1321, 1981 generalized the analysis to an arbitrary number of charge classes.
D. E. Goldman, J. Gen. Physiol. 27:37–60, 1943.
A. L. Hodgkin and B. Katz, J. Physiol. (London) 108:37–77, 1949.
Douglas Junge, Nerve and Muscle Excitation, Second Edition, Sinauer Associates, Sunderland, 1981, 38.
D. C. Eaton, J. M. Russell and A. M. Brown, J. Membrane Biol. 21:353–374, 1975. With kind permission of Springer Science and Business Media.
Resting potential = -58.7 mV; action potential = 43.6 mV.
See Kenneth S. Cole, Physiol. Rev. 45:340–379, 1965, Jarl V. Hägglund, J. Membrane Biol. 10:153–170, 1972, and references cited therein.
The vector quantity D = ∊0 ∊ E is called the electric displacement.
This equation belongs to the quasilinear parabolic class of partial differential equations. It appeared in a 1915 paper by Harry Bateman (Monthly Weather Review 43:163–170) and a 1950 paper by J. M. Burgers (Proc. Kron. Ned. Acad. 53:247–261), and is extensively discussed in Burgers's book, The Nonlinear Diffusion Equation. Asymptotic Solutions and Statistical Problems, D. Reidel, Dordrecht, 1974. Solutions to the Burgers equation are listed in E. R. Benton and G. W. Platzman, Appl. Math. 30:195–212, 1972.
H. R. Leuchtag and H. M. Fishman, in Structure and Function in Excitable Cells, edited by D. C. Chang, I. Tasaki, W. A. Adelman Jr. and H. R. Leuchtag, 415–434, Plenum, New York, 1983.
K. S. Cole, Physiol. Rev. 45:340–379, 1965.
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(2009). Multi-Ion And Transient Electrodiffusion. In: Leuchtag, H.R. (eds) Voltage-Sensitive Ion Channels. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5525-6_8
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